Enumeration of alternating sign triangles using a constant term approach

TL;DR

This paper proves a conjecture on the refined enumeration of alternating sign triangles (ASTs) using a constant term approach, establishing connections to alternating sign matrices, cyclic identities, and loop configurations.

Contribution

It introduces a multivariate generating function for ASTs, proves a conjecture on a specific statistic, and offers an alternative proof for the enumeration of ASTs using operator formulas.

Findings

Confirmed the conjecture on the distribution of a statistic in ASTs

Derived a cyclic rotation identity for ASTs

Connected the generating function to loop configuration formulas

Abstract

Alternating sign triangles (ASTs) have recently been introduced by Ayyer, Behrend and the author, and it was proven that there is the same number of ASTs with n rows as there is of nxn alternating sign matrices (ASMs). We prove a conjecture by Behrend on a refined enumeration of ASTs with respect to a statistic that is shown to have the same distribution as the column of the unique 1 in the top row of an ASM. The proof of the conjecture is based on a certain multivariate generating function of ASTs that takes the positions of the columns with sum 1 (1-columns) into account. We also prove a curious identity on the cyclic rotation of the 1-columns of ASTs. Furthermore, we discuss a relation of our multivariate generating function to a formula of Di Francesco and Zinn-Justin for the number of fully packed loop configurations associated with a given link pattern. The proofs of our results employ the author's operator formula for the number of monotone triangles with prescribed bottom row. This is opposed to the six-vertex model approach that was used by Ayyer, Behrend and the author to enumerate ASTs, and since the refined enumeration implies the unrefined enumeration, the present paper also provides an alternative proof of the enumeration of ASTs.